which amounts to the study of low energy-momentum regime of the theory that is considered as weak perturbation of the flat metric background.
We would study the first and second order term of the perturbation series of the metric. The idea is to consider the metric that can be decomposed into a flat metric with a small perturbation \(h_{\mu \nu }\) , so we will use the linearized perturbation theory of general relativity [1]}.
We just want to find all the geometry quantities that are first order in \(h_{\mu \nu }\) . Up to the first order in \(h_{\mu \nu }\)  next we compute the Christoffel connection
\(\Gamma ^{\rho }_{\mu \nu } = \frac{1}{2} \eta ^{\rho \lambda }(\partial _{\mu }h_{\nu \lambda } + \partial _{\nu }h_{\lambda \mu } - \partial _{\lambda }h_{\mu \nu } )\) 
